Perfect tilings with the generalised triangle in k-graphs

Denote by Tₖ the generalised triangle, a k-uniform hypergraph on vertex set \1,2,,2k-1\ with three edges \1,,k-1,k\,\1,,k-1,k+1\ and ,k+1,,2k-1\. Recently, Bowtell, Kathapurkar, Morrison and Mycroft [arXiv: 2505.05606] established the exact minimum codegree threshold for perfect T₃-tilings in $3$-graphs. In this paper, we extend their result to all k ≥ 3, determining the optimal minimum codegree threshold for perfect Tₖ-tilings in k-graphs. Our proof uses the lattice-based absorption method, as is usual, but develops a unified and effective approach to build transferrals for all uniformities, which is of independent interest. Additionally, we establish an asymptotically tight minimum codegree threshold for a rainbow variant of the problem.